2. Prove that the term $U_1 = \bar{Q}_{11} + \bar{Q}_{22} + 2\bar{Q}_{12}$ is invariant under rotation about the z-axis i.e. independent of angle $\theta$. (10 points)
3. Calculate the stiffness [C], reduced stiffness [Q] and compliance [S] coefficients for AS/3501 graphite-epoxy lamina. The engineering constants for this material are given by; (15 points)
$E_1= 100$ GPa;
$E_2 = E_3 = 20$ GPa,
$\nu_{12} = \nu_{13} = 0.3$;
$G_{23}=G_{12}= G_{13} = 7$ GPa.
4. Using the stiffness and compliance matrix obtained in part 2, calculate the transformed stiffness[$\bar{C}$], transformed reduced stiffness [$\bar{Q}$]and compliance matrix [$\bar{S}$] for rotation of the angle $\theta = 30^\circ$ and $\theta = 60^\circ$ about 3 axis for AS/3501 graphite-epoxy lamina. (20 points)
6. A laminate [60/30/0/30] is prepared using AS/3501 graphite-epoxy lamina (see problem 3 and 4). Calculate the matrices [A], [B] and [D] for the laminate, assuming each layer thickness is 2 mm. (30 points)
